MEDINAH, Ill. — As much as all Ryder Cup players dream about being that guy, they also dread being that guy.

Every one of the 24 players at this week 39th Ryder Cup matches at Medinah wants to be what Graeme McDowell was for his winning European side in 2010 in Wales.

And every one of those players has a pit in his stomach thinking about the possibility of being Hunter Mahan, the American player who chunked a chip like a 20-handicapper in front of the world and lost that decisive 12th singles match to McDowell and the Europeans.

Since that day, McDowell has been hailed as one of the most clutch players on the planet and Mahan has been asked about that horrible moment in practically every interview he has done.

Neither can separate himself from that moment — McDowell hoisted to the sky by jubilant teammates after his win and Mahan inconsolable and in tears while trying to speak to reporters afterward.

By the end of these matches Sunday night, there is likely to be another McDowell and another Mahan — a hero and a goat, to use the simplest of clichés.

“Myself and Hunter Mahan … someone was going to be the hero and someone was going to be the villain that day,’’ McDowell said. “Thankfully, I was able to get the job done, and it was definitely one of the most amazing moments of my career just to be able to share that with 11 great teammates.

“Where will I play on Sunday? Who knows? Part of me would love that opportunity again and part of me would hate it. I’ll take whatever comes.’’

U.S. veteran Jim Furyk called the golf McDowell played in the final holes of that decisive match in 2010 “as clutch golf as I’ve seen or witnessed or personally watched in my career.’’

Of playing the back nine of that match, McDowell said: “I’ve never been so nervous in my life’’ because he realized the fate of his entire team rested on his match.

“You’re just trying not to mess up, [not] lose it for your teammates,’’ he said. “I could have 200,000 spectators watching me, but two of my teammates watching me … begging me to get the job done … there’s something intimidating and very nerve-wracking about that. Their fate rests in your hands at that point.’’

Furyk, who has had some on-course debacles this year, blowing the U.S. Open and the WGC-Bridgestone with final-round gaffes, put the pressure the players face in the best perspective.

“That’s part of what we do for a living,’’ he said. “It’s something that you have to accept. It makes you stronger in the long run. What I went through this year at the U.S. Open, what I went through at Bridgestone, I was definitely not happy about it, but it’s life and it’s sports.’’

Furyk played in the decisive singles match on the Americans’ losing side in 2002 at The Belfry, where he was unable to beat Paul McGinley, and on their winning team in 2008 at Valhalla, where he defeated Miguel Angel Jimenez.

He recalled McGinley halving their singles match on the 18th green to clinch the Ryder Cup for the Europeans and remembered the “empty feeling’’ he had.

“Seeing the place erupt … you feel responsible even though it’s a team event and even though I didn’t lose my match, but that half-point cost us the Ryder Cup and that empty feeling stuck with me,’’ Furyk said. “You don’t know how empty that feeling is until you sit in those shoes.’’

That said, Furyk believes “everyone who’s playing in this tournament would love to be in that position [to win or lose the Ryder Cup]’’ this week.

“You just have to be able to accept the fact that sometimes it turns out good and sometimes it doesn’t,’’ he said.